Studia Scientiarium Mathematicarum Hungarica 34. (1998)

1-3. szám - Berkes I.-Philipp W.: A limit theorem for lacunary series ?f(nkx)

2 I. BERKES and W. PHILIPP then where // is the Lebesgue measure. As an example of Erdős and Fortét (see [8], p. 646) shows, the CLT (1.3) and the LIL (1.4) become generally false if instead of (1.2) we assume only the Hadamard gap condition (1.5) nk+i/nk^q>l {k = 1,2,...). Indeed, let f(x) = cos2irx + cos4nx, nk — 2k — 1. Then, as it is not difficult to show, On the other hand, Kac [7] showed that if / is smooth and nk = 2k then the CLT (1.3) is valid, with the iV(0,1) distribution on the right-hand side replaced by N(0, a2) for some o ^ 0. Thus we see that under (1.5) the asymptotic behaviour of f(nkx) depends not only on the growth speed of (nk), but also on its arithmetic properties. This interesting phenomenon was investigated in detail by Gaposkin [6] who gave a characterization of sequences (nk) satisfying the CLT (1.3) for all sufficiently smooth /. His results imply, e.g., that (1.3) holds if the ratios nk+i/nk are all integers, or if nk+i/nk —> ß where ff is irrational for all positive integers r. For extensions and further limit theorems for f(nkx) see Gaposkin [5], Berkes [1], Berkes and Philipp [3]. It is interesting to note that if we assume only (1.5) then the upper half of the LIL still holds for f(nkx), i.e., for some constant C (see Takahashi [12], Philipp [9]). For further limit theorems for f(nkx) assuming only (1.5) see Berkes [1], and (1.3) (1.4) lim p,{0 ^ x ^ 1 : Y f{nkx) < tVN} = (2n) x^2 I e U^2du JV-> oo L—' / -OO lim sup (2IV log log N)~1/2 ^ f{nkx) = 1 a.e., lim p.{0 re ^ 1: y' f (nkx) < t\/N} — {2n) x^2 I ds I N->oc / / k<N n JL <2l2du and lim N—too sup (2N log log N) 1/2 f(nkx) = \Í2 coi ">0° k<N cos ix x a.e. 1 i/\/2| COS7Ts| (1.6) lim sup (27V log log N)~1/2 N—»oo /(”**•) k<N <C a.e.

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